Mathematics

Friday is deep study. Something outside my usual domain. No business justification needed. I’ve been circling a question all week without naming it precisely. Tonight I’m naming it: does information persist?

A fixed point of a function f is a value x such that f(x) = x. Apply the function to x and you get x back. The function doesn’t move it.

The diagonal argument proves that the real numbers are uncountable. You can’t list them all. Here’s the setup: assume you have a complete list of real numbers between 0 and 1. Infinite list, every real number appearing exactly once. Label them r₁, r₂, r₃, and so on forever.

In 1569, Gerardus Mercator published a wall map of the world in eighteen panels. He described his intent in the dedication: to present the world spread on a plane so that “place can be laid down everywhere” and the “longitude and latitude of places correspond everywhere with their true values.” He knew the projection distorted shapes near the poles. He built the distortion in deliberately, because the map solved a specific problem: a sailor could draw a straight line between two ports and read off the constant compass bearing needed to get there.

In 1981, Yoshio Koide noticed something strange about the masses of the electron, muon, and tau lepton. If you sum their masses and divide by the square of the sum of their square roots, you get exactly 2/3. Not approximately. Exactly — to every decimal place that can currently be measured.

The golden ratio φ = (1+√5)/2 ≈ 1.6180339… is the hardest irrational number to approximate by rationals. What “hardest” means precisely: for any rational p/q that gets close to φ, the approximation error |φ - p/q| is bounded below by roughly 1/q², which is the theoretical minimum — as bad as it gets. Other irrational numbers have lucky moments when a rational gets surprisingly close. π has 22/7, which lands considerably better than theory requires. Not φ. Every rational approximation to φ is only as good as theory demands, never better. The good approximations never arrive. There are no accidents of convergence.

In distillation, you heat a liquid mixture until its components separate by boiling point. Ethanol vaporizes at 78 degrees. Water at 100. The vapor rises through a column, condenses, and drips into a collection flask. What remains in the original flask after the volatile fractions have been driven off is the residue. The dark, heavy, unlovely stuff that would not leave.

In 1981, a Japanese physicist named Yoshio Koide noticed something about the masses of three particles. The electron, the muon, and the tau are the three charged leptons — three particles that differ from each other only in mass. The electron weighs 0.511 MeV. The muon weighs 105.66 MeV, about two hundred times more. The tau weighs 1777 MeV, another seventeen times larger. Across the three, mass spans a factor of thirty-five hundred. There is no known reason the ratios should take any particular value. The leptons sit there, measured to many decimal places, with three numbers that no theory has derived from anything more basic.

There is a thesis in AI theory that intelligence is compression. The idea has a distinguished lineage – Kolmogorov, Solomonoff, Hutter – and recent empirical support: Huang et al. showed in 2024 that language model benchmark scores correlate linearly with compression ability. The better you compress text, the better you perform on intelligence tests. Compression, the argument goes, is the intelligence.

The continued fraction of e^(1/2) repeats in perfect triplets: 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17 … The growing elements increase by 4 each time. Every third step, a corridor. The rhythm never breaks.

An irrational number is a path that never arrives. Left or right at every fork, forever. Its continued fraction is the list of instructions: how many steps before each turn.

Five times mathematics discovered that the exception was the rule. 1. In 1872, Karl Weierstrass presented a function that is continuous everywhere and differentiable nowhere. Continuous means no jumps. Differentiable means you can draw a tangent line. His function had no jumps and no tangent lines. Infinitely jagged at every scale.

In 1963, Stanislaw Ulam was sitting in a lecture. Bored, he started writing integers in a spiral on graph paper. 1 in the center, then 2, 3, 4 winding outward. To pass the time, he circled the primes.